Compressible Viscous Flows in a Symmetric Domain with Complete Slip Boundary

TL;DR

This paper analyzes the stability of compressible viscous flows in symmetric domains with slip boundary conditions, establishing the existence and exponential decay of perturbations towards steady rotating solutions.

Contribution

It introduces a Korn's type inequality tailored for slip boundary conditions and proves the asymptotic stability of rotating solutions in compressible Navier-Stokes flows.

Findings

Existence of steady uniformly rotating solutions.

Perturbations with conserved angular momentum decay exponentially.

Solutions converge to steady states over time.

Abstract

This work is devoted to study the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such scenario the boundary no longer provides friction and therefore the perturbation of angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the asymptotic stability of the uniformly rotating solutions with small angular velocity. In particular, the initial perturbation which preserves the angular momentum would decay exponentially in time and the solution to the Navier-Stokes equations converges to the steady state as time grows up.